In special relativity (SR), there is coordinate time t and proper time τ. Two facts deserve reflection: (1) Clocks measure τ, but the construct t is more common in the equations of physics than natural τ. (2) Cosmology is aware of the Hubble parameter Hθ, but the parameter τ is preferred to θ = 1/Hθ in both SR and general relativity (GR). We show: Euclidean relativity (ER) describes nature exclusively in natural concepts. We apply three steps to make ER work: (1) The new time coordinate is τ. (2) The new parameter is θ. (3) An observer’s reality is a projection from 4D Euclidean space (ES). Because of different concepts, ER does not (!) conflict with SR/GR. All energy moves through ES at the speed c. Absolute ES is experienced as a relative Euclidean spacetime: Each object experiences its 4D motion through ES as its proper time and the other three axes as its proper space. Both the Lorentz factor and gravitational time dilation are recovered in ER. Thus, ER predicts the same relativistic effects as SR/GR. In ER, τ is the length of a 4D Euclidean vector “flow of proper time” τ4D. Clocks with the same τ4D are naturally synchronized. Clocks with different τ4D cannot be synchronized. Gravity makes its comeback as a force. Any acceleration rotates an object’s τ4D and curves its worldline in ES. τ4D is crucial for objects that are very far away or entangled. Information hidden in θ and in τ4D is not available in SR/GR. ER solves 15 mysteries, such as the Hubble tension, the wave–particle duality, and the baryon asymmetry. On top, ER declares cosmic inflation, expanding space, dark energy, and non-locality obsolete.